Evidence of potential axiomatic violations of Prospect Theory (PT), along with the practical solution proposed by rank-dependence models, led Kahneman and Tversky (1992) to develop a more refined model of PT based upon the two elements. Therefore, Cumulative Prospect Theory (CPT) emerged as a more robust framework for analysing decisions under risk and uncertainty. As with the original formulation, prospects are assumed to be evaluated relative to a reference point. Consequently, the normalisation of a reference point to zero enables losses to be defined as any negative outcomes relative to the reference point and gains
as positive outcomes against the same benchmark. Prospects are rank-ordered across a value function such that;
f : = (x1, P1; ….. xk, Pk; xk+1, Pk+1;…; xn, Pn) Equation 12
where; x1 < …… < xk+1 < ……. < xn
As with PT, individuals are assumed to hold different preferences with regard to gains and losses. Negative and positive domains of f can therefore be represented as:
f + : = (0, P1 ∪ … ∪ Pk ; xk+1, Pk+1;…; xn, Pn)
f - : = (x1, P1;…; xk, Pk ; 0; Pk+1 ∪ … ∪ Pn)
where; P1 ∪ … ∪ Pi defines a prospect with an outcome equal to or less than Pi.
The assumption of sign dependence means that independence can only be satisfied within each preference-ranked sign set. The preference weightings with regard to the positive and negative components of a prospect can be represented as;
i+ = W+(Pi ∪ … ∪ Pn) - W+(Pi+1 ∪ … ∪ Pn), for i = k+1, …….., n, and;
i- = W-(P1 ∪ … ∪ Pi) – W-(P1 ∪ … ∪ Pi-1), for i = 1, …….., k
W+ and W- are the separate weighting functions applied to prospective outcomes in the domains of gains and losses relative to the reference point. Sign dependence then implies that decision weights relating to purely positive or negative prospects sum to one. Therefore;
i = [W(Pi ∪ … ∪ Pn) - W(Pi+1 ∪ … ∪ Pn)] = Equation 13
W(P1 ∪ … ∪ Pn) = W(S) = 1
However, such a summation of decision weights across mixed prospects does not necessarily hold meaning that, in such cases, the sum of weights can be less than one (termed subcertainty). Nevertheless, monotonicity is preserved by the strict rank-dependent ordering applied to each domain. It is now possible to represent a prospect under CPT as;
V(f) = i - v(xi) + i + v(xi) Equation 14
where v is a strictly increasing continuous value function with v(0) = 0.
The subjective value of a prospect described by the weighted probability measure p can then be shown as;
V(p) : = ∫ 𝑣(x) (v(F(x))) dx + ∫ 𝑣(x) (-v(1 - F(x))) dx Equation 15
where v is a value function, w is the weighting function and ∫ 𝑑𝑝 is a continuous, increasing cumulative distribution function for all values to x.
3.3.1. Determining Reference Points
While risk under CPT is derived from a combination of three elements: the individual’s basic utility function; unique decision-weights; and loss aversion, the weighting and
reference-dependent value functions are predicated upon the location and nature of the reference point. Determining the reference point is actually critical to the theory as it is that which, along with rank-ordering of prospects within the positive and negative domains, ensures that FOSD is maintained despite subcertainty (the sum of decision weights being less than one with respect to mixed prospects). Yet, CPT, like its predecessor, PT, is silent about how reference points are actually derived.
In his earlier work, referenced at the start of this chapter, Markowitz spoke of “customary wealth” as a potential reference point but had no method of deriving this in the case that it diverged from starting wealth. For their part, Tversky and Kahneman (1991) also considered that a reference point would typically correspond with a “current wealth position”, although suggested that this could be modified by expectations or aspirations. In general, therefore, reference points were largely assumed to default to a status quo position or were considered to be psychological in nature. This lack of clarity clearly hinders empirical testing of a reference- dependent theory such as CPT as it allows too much latitude in terms of specification; the theory might be adapted to fit most observed outcomes.
In practice, there is ample scope for decision-makers to set single or even multiple reference points and to shift or modify them over time. Similarly, while CPT appears to imply that a status quo reference point is a certain alternative to risky prospects, there is no reason to assume that they cannot be stochastic. Indeed, stochastic reference points are assumed when those reference points involve expectations (Koszegi & Rabin, 2006). Since expectations can be assumed to involve uncertainty, prospects can then be considered as elements within lotteries offering an array of potential payoffs. The comparison can then be made with an alternative lottery which provides the relative reference points. Therefore, if we define a
lottery, L, offering N possible outcomes with associated probabilities, p; L = (x1, p1, …. xn, pn),
subject to ∑ 𝑝i = 1, there exists a reference lottery, R, with M potential outcomes; R = (r1,
q1, …. Rm, qm), subject to ∑ 𝑞j = 1. Decision-makers then evaluate the prospects contained
in L against those found in R in the following manner;
U(L | R) = ∑ 𝑝i [u(xi) + v(xi | R)] Equation 16
where, u(xi) is the expected utility from outcome xi and v(xi | R) measures the utility
(subjective value) associated with the gain or loss from outcome xi measured relative to
the outcome distribution in R.
Loomes and Sugden (1986), modelling disappointment aversion, applied such a model whereby xi was assumed to be compared with a summary statistic from R such that;
v(xi | R) = (u(xi) – ( ∑ 𝑞, u(xj)) Equation 17
Koszegi & Rabin (2006) proposed a more general model in which each xi is compared with
every prospect in R;
v(xi | R) = ∑ 𝑞j u(xi) – u(xj) Equation 18
Therefore, xi > xj represents a gain over all such elements in R, while; xi < xj represents a loss.
The assumption is that lottery L is then viewed on the basis of an assessment of all gains and losses relative to the elements in R.
Note that, while both approaches combine classic utility from consumption with utility associated with a gain or loss, they make different predictions based upon an individual’s preferences relating to gains and losses. This becomes apparent if, for example, there is an increase in risk associated with R which does not affect the decision-maker’s utility with regard to R. Assuming the prospect comparisons in Equation 17, a change in risk in R should have no impact on overall expected behaviour. However, under the broader formulation of comparisons in Equation 20, an increase in the risk of R should make the decision-maker correspondingly more willing to accept risk in L, creating a type of endowment effect with respect to risk.
Koszegi & Rabin further suggested that reference points are likely to be transient in nature, shifting over time and when actual choices are made. Therefore, given the lottery L is actually chosen, at some point that lottery replaces R as the reference benchmark against which actual outcomes are judged in accordance with gains or losses relative to expectations for L. Koszegi and Rabin (2007) applied this concept more formally defining a “choice- acclimatising personal equilibrium”. Essentially, if a decision-maker commits to a choice, such as lottery L, well in advance of realising the actual outcome, expectations about L will become established as the new reference point by the time the outcome becomes known. He and Strub (2019) extended this within a mental-adjustment model with loss version, finding shifts in both exogenous and endogenous reference points over time.
Approaching the issue of how such expectations are formed, the process can be slightly simplified if we assume that expectations are either exogenous or endogenous to the current choice. In the examples above, the primary reference set defined by R was exogenous to choice L. If reference points shift after choices have been made, such that R is replaced by L,
expectations are then endogenous to choice L. From Equation 16, we see that the individual’s utility would then be given by; U(L|L). Rational behaviour would then require selection of the lottery which maximises that term.
There is ample scope to assign psychological traits with regard to the treatment of reference points. For example, disappointment aversion can apply as decision-makers are likely to develop prior expectations with regard to a lottery which may not be realised. If we apply the two alternative methods of comparison described above in relation to disappointment aversion (DA) and the Koszegi & Rabin (KR) formulation, we find the following interpretations with regard to gain-loss utility;
DA: U(L|L) = ∑ 𝑝i [u(xi) + u(xi) – ∑ 𝑝j u(xj)] Equation 19
KR: U(L|L) = ∑ 𝑝i [u(xi) + ∑ 𝑝j u(xi) - u(xj))]
It may be noted that the KR model implies that deviations from EU are based upon subjective assessments of gains versus losses, although there is no equivalent of a value function to specify this process. Nevertheless, there is implied information regarding a prospect’s ranking as, under this formulation, an outcome’s rank depends upon the number of gain/loss comparisons made with the reference set. This has some obvious similarities with rank-dependent probability weighting models which transform the cumulative probability distribution.
Bhatia and Golman (2015) presented a model of reference dependence extending the concept that the choice of reference points is influenced by particular attributes of those reference points. Behaviours such as the endowment effect and status quo bias, for example,
have been explained on the basis of attention towards attributes or reference points (Ashby, Dickert & Glockner, 2012); the decision-maker then places a greater weight on key attributes when determining preferences. Therefore, while reference points are not assumed to affect perceptions regarding gains and losses, they do impact choice through a search for, and comparison of, similar attributes. As a result, changing reference points might result in changes in the weights assigned to particular attributes, perhaps modifying choices. The result is a model of attention-biased utility based upon subjective valuations of key attributes. On this basis, and assuming that attributes are considered to be mutually independent, attribute- biased utility maximisation can simply be defined in the standard way as;
U(x) = ∑ 𝑉i (xi) Equation 20
where V is the decision-makers subjective value function with Vi (xi) then denoting the
valuations of attribute i with regard to option x.
The weighting of specific attributes is accommodated by applying a non-negative, strictly increasing attention function, a = a(r) which defines the individual’s attention weight given a particular reference point r. Choice is then made according to the following weighted attribute utility function;
U(x|r) = ∑ 𝑎i (ri) - ai(ri) · Vi(ri) Equation 21
Note that no assumptions are made about the nature of the value function, V; utility maps according the value function regardless of the reference point. In addition, there is no assumed divergence in terms of the assessment of gains or losses relative to a particular reference point. Instead, preferences are derived in accordance with a prospect’s primary
attributes defined according to the subjecting weighting function. The attribute utility model is consistent with evidence that salient choice options affect an individual’s attention to choice characteristics (Pachur & Scheibehenne, 2012) and hence preferences. The model is therefore capable of explaining a number of observed behavioural anomalies, including a reversal of the endowment effect for negative attributes and the strengthening of the effect for more highly weighted attributes.
Early approaches to modelling multiple reference points tended to assumed that decision makers reduced these points to a single composite (Olson, Roese & Zanna, 1996), although the proposition has been challenged by Ordóñez, Connolly and Coughlan (2000). A typical assumption is that decision-makers applying multiple reference points impose minimum goals which they strive to attain. This has been found to elicit behaviour contradictory to Prospect Theory to the extent that risk aversion often does not appear once performance exceeds the status quo (Sullivan & Kida, 1995). Thus, while business managers have been found to exhibit risk aversion when confronted with the possibility of losses sufficient to return performance to the prior status quo, they have also been found to be risk seeking in pursuit of higher goals once the status quo had been sufficiently exceeded. Sullivan and Kida concluded that each of the multiple goals could exert some influence of behaviour simultaneously.
March and Shapira (1992) proposed a variable risk preference model such that decision- makers could switch attention between multiple reference points and goals. However, they assumed that the influence of each was mutually exclusive, implying that decision-makers adopt either a fully risk averse, preservation strategy or an active risk seeking strategy in pursuit of goals. This approach has some similarities with Lopes’s “surviving and thriving” models (Lopes, 1987), although the latter focussed more on transposed decision weights rather
than reference points. The effect of multiple reference points on strategic behaviour has been considered with regard to interactive, mainly bilateral, negotiations (Neale & Bazerman, 1991; Neale, Huber & Northcraft, 1987). In relation to real estate purchases, White, Valley,
Bazerman, Neale & Peck (1994) found that reservation prices tended to act as dominant reference points along with the maximum willingness to pay of the buyer. Kristensen and Garling (2000) found that initial offers could affect prior reference points, leading to
modification in anchors. Within this process, a distinction is normally drawn between anchors and reference points (Kahneman, 1992). Thus, anchors are perceived to exist in terms of offer and counteroffer levels whereas reference points determine whether those offers and
counteroffers are perceived as gains and losses. A reference point may therefore influence counteroffers depending on how the new anchor point is perceived. Kristensen and Garling (1997) found that proposed selling prices acted as anchor points for basing counteroffers.
A further interactive approach to multiple reference points was provided by Wang and Johnson (2009) using a model with three separate reference points; the status quo, a coalesced multiple reference point and aspirational goals. By cross-referencing outcomes with these triple “areas of outcome”, decision-makers could experience success (gain) or failure (loss). Wang and Johnson argue that each of the outcome regions (staying above the status quo, exceeding reference points or exceeding a goal) could have different decision weights associated with each. Therefore, a salesperson exceeding a base target level may derive no increase in overall utility in the event that such performance falls short of a higher goal which would have triggered a financial bonus. This conflicts with Prospect Theory’s assumption regarding relative insensitivity to small changes in outcomes since, in this example, a
potentially modest increase in sales performance could potentially result in a tangibly superior outcome by triggering the bonus threshold. In such cases, great importance is likely to be
attached to achieving that small additional gain. It might therefore be assumed that such reference points will reflect critical levels of perceived utility outcomes, in which case the overall value function is likely to exhibit a greater slope in the neighbourhood of such critical reference points rather than the status quo point. Consequently, decision-makers might be expected to accept higher risk gambles in the neighbourhood of such reference points if the risky prospect offers the chance of exceeding a certain threshold.
The tri-reference model of Wang and Johnson was tested in two experiments by Koop and Johnson (2010). In the first experiment, participants were presented with a number of risky gambles within the broader context of three reference points; a minimum requirement (MR), the status quo (SQ) and a goal (G). Based upon prospective payoffs, various return outcomes were possible with respect to individual reference points. Consequently, a general outcome space could be mapped as shown in Figure 5.
Table 7. Outcomes from risky gambles relative to objective reference points ‘mapped’ into a general outcome space.
Outcome of Gamble x Classification of Outcome
x < MR Failure
MR ≤ x < SQ Loss
SQ < x < G Gain
G ≤ x Success
A total of 12 notional risky gamble pairs were then prepared for presentation, as shown in Table 8. In each of the gamble pairs, explicit values (expressed in lira) for reference points were assigned such that MR = 1000, SQ = 2000 and G = 4000. In each case, the probability of outcomes associated with each of the risky gambles was declared to be equal, meaning that Pr(a1) = Pr(a2) = p = .5. It will be noted that each return pair, a1 and b1, resides in the same
point). Therefore, in the case of Pairs 1, 5 and 9, the associated outcomes for a1 and b1
represent failure (outcomes below MR). The a2and b2binary outcomes were set so that they
resided in adjacent outcome spaces thereby straddling a defined reference point. For example, in the case of Pair 9, a2 represents a gain (SQ < x(a2) < G) while b2represents success (x(b2) >
G), using the previously defined outcome descriptors.
Table 8. Paired risky gambles with common outcomes relative to specified reference points.
Pair Number
Gamble A Gamble A Common outcome Reference point
a1 a2 b1 b2 (a1, b1) involved (a2, b2)
1 940 960 580 1220 Failure MR 2 1880 920 1600 1100 Loss MR 3 3840 760 3300 1200 Gain MR 4 4900 600 4220 1180 Success MR 5 880 1920 620 2080 Failure SQ 6 1720 1880 1420 2080 Loss SQ 7 3700 1800 2820 2580 Gain SQ 8 4600 1800 4200 2100 Success SQ 9 760 3860 400 4120 Failure G 10 1800 3700 1160 4240 Loss G 11 3100 3600 2360 4240 Gain G 12 5060 3680 4400 4240 Success G
Note: The data shows pairs of gamble outcomes which are assumed to have equal probabilities (.5) of success. The gambles are notionally evaluated against pre-defined, fixed reference points representing a minimum return requirement (MR), the status quo (SQ) or a gain (G). Paired outcomes are then categorised on the basis of outcomes versus the various reference points such that an outcome, x, below MR is defined as “Failure”, x > SQ denotes “Loss”, x > SQ < G defines “Gain”, x > G = “Success”.
By varying the values of a1 and b1while maintaining the reference points straddled by a2 and
b2, three sets of gamble pairs (4 x 3) were presented to each subject representing differing
possible outcomes across the entire outcome space. In the case of each gamble, the values of a1, b1, a2 and b2 were chosen so that A always had a higher expected return than B, whereas B
offered the prospect of a superior functional outcome as it allowed the possibility of exceeding a specific reference point.
In order to test the strength of any reference point effect on behaviour, two conditions were defined. In the first, strong (certain) condition, each of the three reference points was disclosed to participants as a single value. In the second, weak (uncertain) condition, SQ remained fixed while MR and G were described in terms of symmetrical probability
distributions around a mean equal to the value declared in the strong condition. Participants were incentivised in terms of both instant monetary reward, based upon achieving certain benchmarks, and the prospect of entry into a future bonus draw with its own payoff.
Results from the experiment indicated that participants generally preferred gambles associated with achieving reference points as opposed to seeking the highest expected payoff. This was most marked under the strong condition where reference dependent gambles were chosen in preference to higher payoff gambles over 78% of the time. The same propensity for reference point dependence was also found for the weak condition, although the level of significance was lower than that found in the strong condition. The effect was, however, particularly strong around MR. Preferences for gambles around SQ were equally significant for both the strong and weak conditions. Unlike MR and G, SQ was fixed under both conditions and therefore not subject to the degree of uncertainty found within the weak condition with regard to other reference points, defined by their distribution of outcomes rather than a single value. It would appear, therefore, that the introduction of a degree of uncertainty had some impact upon strength of preferences over and above simple risk aversion around reference points.
The effect of multiple reference point dependence differentiable from simple risk aversion, expected utility and SQ dependence implied by Prospect Theory was investigated
explicitly in a second experiment. Koop and Johnson employed a similar methodology in terms of selective gambles defined precisely across a reference dependent outcome space. A number of the gambles applied in experiment 1 were restated in experiment 2 in order to create a more effective delineation of expected outcomes based upon the various possible objective functions (reference dependence, risk aversion, expected utility and prospect theory). The new values used are shown in Table 9.
Table 9. Paired risky gambles with common outcomes relative to specified reference points used in experiment 2.
Note: Modified data used in experiment 2 designed to test behaviour in relation to behavioural objectives (reference dependent, risk aversion, expected utility and prospect theory).
The values assigned to explicit reference points, denominated in lira, were; MR = 1000, SQ = 2500 and G = 4000. The common outcome and reference points were as described in the prior